Shami Chakrabarti, the director of Liberty, today threatened to sue culture secretary Andy Burnham over a comment he made linking her to David Davis.

She described it as a "smear" and said she would take him to court if he did not apologise for the remark, which he made in an interview with Progress magazine about her links with the former shadow home secretary.

In the interview, Burnham said that it was "curious" that Chakrabarti was supporting Davis - who resigned to fight a byelection over his opposition to the government's plans for 42-day pre-charge detention - given Davis's rightwing stance on issues like capital punishment.

Burnham said: "To people who get seduced by Tory talk of how liberal they are, I find something very curious in the man who was, and still is I believe, an exponent of capital punishment having late-night, hand-wringing, heart-melting phone calls with Shami Chakrabarti."

Chakrabarti, and Davis, who are both married to other people, were both said to be furious.

On Wednesday night, following the publication of the interview, a spokeswoman for Burnham denied that there was any suggestion of impropriety.

"It was a light-hearted comment about the former shadow home secretary's political journey, byelection political knockabout and nothing else. Nothing more should be read into it and no personal offence was intended to Shami Chakrabarti," the spokeswoman said.

But this did not satisfy Chakrabarti, who today issued her legal threat in a letter to Burnham that was also released to the media.

She said she found his remark "curious, coming as it does from a cabinet minister; let alone someone with a partner and family of his own".

She went on: "By your comments you debase not only a great office of state, but the vital debate about fundamental rights and freedoms in this country. Indeed you seem reluctant to engage in that debate except in this tawdry fashion.

"I look forward to your written apology as I'm sure does Mrs Davis. If on the other hand you choose to continue down the path of innuendo and attempted character assassination, you will find that the privileged legal protection of the parliament chamber does not extend to slurs made in the wider public domain. The fruits of any legal action will of course go to Liberty (the National Council for Civil Liberties)."

In response to the letter, Burnham's office issued a further clarification.

The statement said: "Andy Burnham was making a political point about David Davis's inconsistent views on capital punishment and civil liberties.

"An interpretation has been placed on Andy's remarks that he did not intend. His comments related to politics and nothing else. He regrets if any personal offence has been caused."

But this afternoon Liberty said that on its own, this response would not be enough for Chakrabarti to withdraw her threat to sue.

"We are still awaiting a response," a Liberty spokeswoman said. "Simply putting out a statement, not even to us directly, is not a real apology."

Asked about the situation at the afternoon briefing for lobby journalists, the prime minister's spokesman said Gordon Brown had full confidence in Burnham.

In a statement issued yesterday, Davis said: "Labour has lost the argument over the erosion of British freedoms. "While Gordon Brown cowers in Downing Street, his henchmen are out and about to attack me personally rather than engage in rational debate.

"Labour has now resorted to personal smears and lies rather than make its case for 42-day detention and for the other illiberal measures it has taken."